$\lim_{x\to \frac{\pi}{6}}\cot(x)=?$ Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\sqrt{3}}{3}$ (Choice B) B $\dfrac{\sqrt{3}}{2}$ (Choice C) C $\sqrt{3}$ (Choice D) D The limit doesn't exist.
Explanation: $\cot(x)$ is continuous on all points in its domain. Therefore, if $x=\dfrac{\pi}{6}$ is within the domain of $\cot(x)$, we can find $\lim_{x\to \frac{\pi}{6}}\cot(x)$ by direct substitution. $x=\dfrac{\pi}{6}$ is indeed in the domain of $\cot(x)$ : $\begin{aligned} \cot\left(\dfrac{\pi}{6}\right)&=\dfrac{\cos\left(\dfrac{\pi}{6}\right)}{\sin\left(\dfrac{\pi}{6}\right)} \\\\ &=\dfrac{\left(\dfrac{\sqrt{3}}{2} \right)}{\left( \dfrac{1}{2} \right)} \\\\ &=\sqrt{3} \end{aligned}$ $\lim_{x\to \frac{\pi}{6}}\cot(x)=\sqrt{3}$